Optimal. Leaf size=73 \[ \frac {\cos ^7(c+d x)}{7 a^3 d}-\frac {\cos ^6(c+d x)}{2 a^3 d}+\frac {3 \cos ^5(c+d x)}{5 a^3 d}-\frac {\cos ^4(c+d x)}{4 a^3 d} \]
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Rubi [A] time = 0.16, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3872, 2836, 12, 43} \[ \frac {\cos ^7(c+d x)}{7 a^3 d}-\frac {\cos ^6(c+d x)}{2 a^3 d}+\frac {3 \cos ^5(c+d x)}{5 a^3 d}-\frac {\cos ^4(c+d x)}{4 a^3 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2836
Rule 3872
Rubi steps
\begin {align*} \int \frac {\sin ^7(c+d x)}{(a+a \sec (c+d x))^3} \, dx &=-\int \frac {\cos ^3(c+d x) \sin ^7(c+d x)}{(-a-a \cos (c+d x))^3} \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {(-a-x)^3 x^3}{a^3} \, dx,x,-a \cos (c+d x)\right )}{a^7 d}\\ &=\frac {\operatorname {Subst}\left (\int (-a-x)^3 x^3 \, dx,x,-a \cos (c+d x)\right )}{a^{10} d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-a^3 x^3-3 a^2 x^4-3 a x^5-x^6\right ) \, dx,x,-a \cos (c+d x)\right )}{a^{10} d}\\ &=-\frac {\cos ^4(c+d x)}{4 a^3 d}+\frac {3 \cos ^5(c+d x)}{5 a^3 d}-\frac {\cos ^6(c+d x)}{2 a^3 d}+\frac {\cos ^7(c+d x)}{7 a^3 d}\\ \end {align*}
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Mathematica [A] time = 1.76, size = 80, normalized size = 1.10 \[ \frac {4060 \cos (c+d x)-3220 \cos (2 (c+d x))+2100 \cos (3 (c+d x))-1120 \cos (4 (c+d x))+476 \cos (5 (c+d x))-140 \cos (6 (c+d x))+20 \cos (7 (c+d x))-2421}{8960 a^3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 49, normalized size = 0.67 \[ \frac {20 \, \cos \left (d x + c\right )^{7} - 70 \, \cos \left (d x + c\right )^{6} + 84 \, \cos \left (d x + c\right )^{5} - 35 \, \cos \left (d x + c\right )^{4}}{140 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.27, size = 163, normalized size = 2.23 \[ \frac {4 \, {\left (\frac {91 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {273 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {455 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {490 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {210 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {140 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 13\right )}}{35 \, a^{3} d {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.64, size = 50, normalized size = 0.68 \[ -\frac {\frac {1}{2 \sec \left (d x +c \right )^{6}}-\frac {1}{7 \sec \left (d x +c \right )^{7}}+\frac {1}{4 \sec \left (d x +c \right )^{4}}-\frac {3}{5 \sec \left (d x +c \right )^{5}}}{d \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 49, normalized size = 0.67 \[ \frac {20 \, \cos \left (d x + c\right )^{7} - 70 \, \cos \left (d x + c\right )^{6} + 84 \, \cos \left (d x + c\right )^{5} - 35 \, \cos \left (d x + c\right )^{4}}{140 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 58, normalized size = 0.79 \[ -\frac {\frac {{\cos \left (c+d\,x\right )}^4}{4\,a^3}-\frac {3\,{\cos \left (c+d\,x\right )}^5}{5\,a^3}+\frac {{\cos \left (c+d\,x\right )}^6}{2\,a^3}-\frac {{\cos \left (c+d\,x\right )}^7}{7\,a^3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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